Image Processing Reference
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Fig. 20. FDM input spectrum for selection and separation by two-channel directional filter
(DF)
and two output signals Göckler (1996a). For convenience, we map the original odd indices
c
∈{
1, 3, 5, 7
}
of the COHBF centre frequencies to natural numbers as defined by
f n
8 ,
=(
+
) ·
∈{
}
f o
2 o
1
o
0, 1, 2, 3
(38)
for subsequent use throughout Section 3.
Section 3 is organized as follows: In Subsection 3.1, we detail the statement of the problem,
and recall the major properties of COHBF needed for our DF investigations. In the
main Subsection 3.2, we present and compare two different approaches to implement the
outlined LP DF for signal separation with selectable centre frequencies: i ) A four-channel
uniform complex-modulated FDMUX filter bank undercritically decimating by two, where
the respective undesired two output signals are discarded, and ii ) a synergetic connection of
two COHBF that share common multipliers and exploit coefficient symmetry for minimum
computation. In Subsection 3.3, we apply the transposition rules of [Göckler & Groth (2004)]
to derive the dual DF for signal combination (FDM multiplexing). Finally, we draw some
further conclusions in Subsection 3.4.
3.1 Statement of the DF problem
Given a uniform complex-valued FDM signal composed of up to four independent user
signals s o (
e j Ω )
) ←→
S o (
= {
}
, according to (38), as depicted
in Fig. 20, the DF shall extract any freely selectable two out of the four user signals of
the FDM input spectrum, and provide them at the two DF output ports separately and
decimated by two:
kT n
centred at f o , o
0, 1, 2, 3
e j Ω ( d )
s o (
)
=
s o (
mT d ) ←→
S o (
)
=
=
2 T n . R l
that complex-valued time-domain signals and spectrally transformed versions thereof are
indicated by underlining.
Efficient signal separation and decimation is conceptually achieved by combining two
COHBF with their differing passbands centred according to (38), where o
2 kT n
:
; T d
1/ f d
∈{
}
,
along with twofold polyphase decomposition of the respective filter impulse responses
[Göckler & Damjanovic (2006a); Göckler & Groth (2004)]. All COHBF are frequency-shifted
versions of a real zero-phase (ZP) lowpass HBF prototype with symmetric impulse response
h
0, 1, 2, 3
e j Ω ) R
according to Subsection 2.1.1, as depicted in Fig. 21(a)
as ZP HBF frequency response [Milic (2009); Mitra & Kaiser (1993)]. A frequency domain
representation of a possible DF setting (choice of COHBF centre frequencies o
(
k
)=
h k =
h
←→
H 0 (
k
∈{
0, 2
}
)is
shown in Fig. 21(b), and Figs.21(c,d) present the output spectra at port I ( o
=
0) and port II
( o
f n /2.
A COHBF is derived from a real HBF (9) by applying the frequency shift operation in the time
domain by modulating a complex carrier z o =
=
2), respectively, related to the reduced sampling rate f d =
e j k ( 2 o + 1 ) 4 of a frequency prescribed
e j2 π kf o / f n
=
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